That very much depends on the PDF. For some distributions, the mean itself is necessary and sufficient (like the exponential), for others the mean and standard deviation are necessary and sufficient (the Gaussian), and for still others the distribution is not parametrized by the mean and std. If this is Gaussian, then
I'll assume mu is the mean and sigma the standard deviation.

By assuming a Gaussian distribution, essentially all the probability is the mean plus/minus 3 standard deviations, so I just put an extra one (sigma) in there as a cushion. Any Gaussian function really has infinite support, but for all intents and purposes, you can characterize the PDF by just considering that interval.